We aim to slow a supersonic, molecular beam of $^{11}$BH using a Zeeman slower and subsequently cool the molecules to sub-millikelvin temperatures in a magneto-optical trap. Most molecules are not suitable for direct laser cooling because the presence of rotational and vibrational degrees of freedom means there is no closed-cycle transition which is necessary to scatter a large number of photons. As was pointed out by Di~Rosa\footnote{M.~D.~Di~Rosa, \textit{The European Physical Journal D}~\underline{\textbf{31}}, 395, 2004} there exists a class of molecules for which the excitation of vibrational modes is suppressed due to highly diagonal Franck-Condon factors. Furthermore, Stuhl et al.\footnote{B.~K.~Stuhl et al., \textit{Physical Review Letters}~\underline{\textbf{101}}, 243002, 2008} showed that angular momentum selection rules can be used to suppress leakage to undesired rotational states. Here we present a measurement of the radiative branching ratios of the $A^1\Pi\rightarrow X^1\Sigma$ transition in $^{11}$BH - a necessary step towards subsequent laser cooling experiments. We also perform high-resolution mm-wave spectroscopy of the $J'=1\leftarrow J=0$ rotational transition in the $X^1\Sigma (v=0)$ state near 708~GHz. From this measurement we derive new, accurate hyper fine constants and compare these to theoretical descriptions. The measured branching ratios suggest that it is possible to laser cool $^{11}$BH molecules close to the recoil temperature of 4~$\mu$K using three laser frequencies only.